Koenigs

Koenigs refers to a family of mathematical concepts and related theorems, primarily associated with the work of German mathematician Gotthold Eisenstein and later expanded upon by others. The most prominent of these is the Koenigs graph, a specific type of infinite graph constructed using integer coordinates.

The Koenigs graph can be visualized as a grid where vertices are points with integer coordinates (x, y) in a plane. Edges connect two vertices if the Euclidean distance between them is a prime number. This construction leads to fascinating properties, particularly regarding its connectivity and Hamiltonian paths.

One significant result related to the Koenigs graph is the conjecture that it is Hamiltonian, meaning it contains a Hamiltonian cycle, which is a path that visits every vertex exactly once and returns to the starting vertex. While this conjecture remains an open problem, research has explored related concepts like Hamiltonian paths and cycles in variations of the Koenigs graph.

The term "Koenigs" can also appear in other mathematical contexts, often related to graph theory or number theory, but the graph construction is its most widely recognized association. The study of Koenigs graphs and related problems contributes to the broader understanding of infinite graphs and their structural properties.